Positivity violations of the density operator in the Caldeira-Leggett master equation

TL;DR

This paper investigates the mathematical consistency of the Caldeira-Leggett master equation, revealing conditions under which the density operator's positivity is violated, with both analytical and numerical insights.

Contribution

It provides new analytical solutions and numerical evidence of positivity violations in the Caldeira-Leggett master equation, expanding understanding of its limitations.

Findings

Positivity violations occur under certain conditions.

Analytical solutions via the method of characteristics.

Numerical results show violations of uncertainty relations.

Abstract

The Caldeira-Leggett master equation as an example of Markovian master equation without Lindblad form is investigated for mathematical consistency. We explore situations both analytically and numerically where the positivity violations of the density operator occur. We reinforce some known knowledge about this problem but also find new surprising cases. Our analytical results are based on the full solution of the Caldeira-Leggett master equation obtained via the method of characteristics. The preservation of positivity is mainly investigated with the help of the density operator's purity and we give also some numerical results about the violation of the Robertson-Schr\"odinger uncertainty relation.